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Mechanical Systems and Signal Processing (2003) 17(2), 273283
doi:10.1006/mssp.2002.1531, available online at http://www.idealibrary.com on
FAULTDETECTIONINGEARDRIVESWITH
NON-STATIONARYROTATIONALSPEED}
PART II:THETIME^QUEFRENCYAPPROACH
G. Meltzer
Technical Diagnostics, Department of Energy Machines and Machine Laboratory, Faculty ofMechanical Engineering, Dresden University of Technolog y, D-01062 Dresden, Germany.
E-mail: [email protected]
an d
Yu.Ye. Ivanov
ZF Friedrichshafen, Germany
(Received 1 February 1999, accepted in revised form 23 September 2002)
This paper deals with the recognition of faults in toothing during non-stationary start upand run down of gear drives. In the first part, this task was solved by means of the timefrequency analysis. A planetary gear was used as a case study. Part II contains a newapproach using the timequefrency analysis. The same example was successfully subjectedin this procedure.
# 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The appearance of sidebands in the frequency spectrum of measured structure-borne noise
of gear cases is, according to our present knowledge, a sure symptom for faults in
toothing. These sidebands are generated by the amplitude modulation or frequency
modulation, resp. of the meshing vibrations, as a result of meshing of faulty teeth [1,2].
The sideband analysis was executed by means of the classical Fourier analysis as well
as}because of the non-stationarity of rotational speed during measuring}also by means
of the timefrequency analysis. Depending on the chosen time and frequency resolutionsthe contour plots of timefrequency distribution (TFD) shows more wavy lines
(representing the instantaneous frequency), dots on straight lines (representing the
instantaneous amplitude) or parallel straight frequency lines (representing the meshing
frequency and sidebands). The last-mentioned analysis was used in Part I of our paper [3]
and tested as an approach for fault diagnostics in gearboxes.
The ordering spectrum as the result of Fourier transformation of the angle-equidistant
sampled measured signal shows the order of teeth meshing (this is equal to the number of
teeth multiplied with the rotational frequency of the shaft) accompanied by some
sidebands around the meshing frequency. In the plot of time-ordering distribution (TOD),
these sidebands marked perpendicularly orientated straight lines. The spacing of sidebandsto the meshing order is a symptom for the location of the fault. Therefore, these distances
had to be determined very carefully. For doing this, we applied the well-known tool of
cepstral analysis. A cepstrum is the inverse Fourier-transformed logarithm of the
belonging-to spectrum [1, 2]. Periodically located sidebands in the spectrum are
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characterised by particular peaks in the cepstrum. The simultaneous appearance of two
peaks in the cepstrum with quefrencies correlates to the rotational speed of central gear
and annulus of second planetary detected toothing faults or damages at the central gear as
well as at the annulus [3].
A similar tool for exact recognition and following the precise identification of sideband
spacings out from TFD in the case of non-stationary rotational speed was not availableuntil now. But there exists a necessity for this, in order to solve the following problem:
How can the existence of gear fault be detected in the case of continuously changing
rotational speed? In gear drives, the sideband distance changes with the rotational speed of
the damaged wheels [1, 4]. Any unsteady rotational speed consequently generates time-
dependent sideband spacings. In this case, the application of the ordinary Fourier
cepstrum is not suitable because of the expected smearing over of the cepstral peaks.
Therefore, it seems to be unable to identify the toothing faults outgoing from time-
equidistant sampled measuring values [3]. On the other side, the TFD has shown these
sidebands at the planetary set 2 clearly enough.
To avoid the implementation of a phase angle encoder (please remember: sampling insmall angle intervals would be an easy approach for solving our task, but it is connected
with more hardware expense), we have to find an until-now-missed method for
transformation of the TFD by a time quefrency analysis (TQA) into a timequefrency
distribution (TQD) in order to identify the sideband distances also quantitatively. Our
target was now to create and to apply such a new tool on the basis of the time-equidistant
sampled measuring data. We have done this successfully}about the result will be reported
furthermore.
2. THE METHOD OF TIMEQUEFRENCY ANALYSIS
We are concerned here with quadratic timefrequency distributions TFDXt;o of
Cohen class which fulfil the condition
Xjoj j2
Z11
TFDxt;o dt 1
withjXjoj2 as the square of magnitude of the Fourier-transformed signal [57].
Equation (1) shows that the Fourier powerspectrum can be reconstructed out of the
TFD of the belonging-to signal. By executing the logarithm and subsequently the inverse
Fourier transformation of both sides of equation (1) we get
Cxt
Z11
ln
Z11
TFDxt;o dt
expjot do 2
with CXt as the ordinary Fourier power cepstrum of xt:
Cxt
Z11
ln Xjoj j2 expjot do: 3
Equations (2) and (3) show that on the premise of equation (1) also the Fourier cepstrum
can be reconstructed from the timefrequency distribution TFDXt;o:
We want to define a new transformation similar to equation (3), in which the Fourier-
power spectrum shall be replaced by the time-dependent power spectrum or the time
frequency distribution TFDXt;o:
CTQDxt; t
Z11
ln TFDxt;o expjot do: 4
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The left side of equation (4) is the inverse Fourier-transformed of logarithm of the time
frequency distribution TFDXt;o: It depends on two variables: the time t and the
quefrency t: Therefore, it can be named the TQD.
The inverse Fourier transformation (3) is used for detection of periodic components of
the spectrum jXjoj2 of signal xt: The inverse Fourier transformation (4) gives the same
insight, but additionally it shows these periodicities in the frequency domain x asfunctions of time. Because of it, the new created function CTQDXt; t can be coordinated
to the class of cepstral functions as a TQD. It shows the momentary cepstrum of xt;
which depends generally on time and which can be favourably used for a quantitative
identification of transient periodicities on frequency in the spectrum ofxt: It can be also
averaged over the time to the time-averaged cepstrum. The result is
Gxt
Z11
CTQDxt; t dt: 5
Equation (5) is similar to equation (2): both are time-independent cepstra. But
nevertheless, they are not identical: CXt=GXt: This can be proved by substitution ofequation (4) into (5) with the result (6) and following comparison with equation (2):
Gxt
Z11
Z11
ln TFDxt;o expjot do
dt: 6
We lost the identity by changing the sequence of the operations of logarithm and
integration in equations (2) and (6), resp. The conclusion is that the Fourier power
cepstrum CXt cannot be reconstructed outgoing from the special form of TQD, which is
formulated in equation (5) as CTQDXt; t:
The mutual relations between spectrum, cepstrum, TFD and TQD are shown in Fig. 1.
Now something about the properties of the new created class of TQD.It is well known that the Fourier power cepstrum CXt is a function of quefrency t with
real function quantities only. This is caused by the circumstance that the Fourier spectrum
jXjoj2 is an even function ofx. Also the mostly used representants of Cohen class, the
WignerVille distribution (WVD) and the ChoiWilliams distribution (CWD) are even
functions of o: TFDXt;o TFDXt; o [6]. But nevertheless, the timequefrency
distribution CTQDXt; t is generally a function with complex function quantities. This is
caused by the logarithmic operation of the TFD in equation (4), which provides not
always positive function quantities. This is marked by the first capital C of the
abbreviation CTQD. In the case of only positive function quantities of TFD TFDXt;o
> 0 for arbitrary t; o) we get a TQD with only real function quantities.Only few of TFD of Cohen class possess the so-called positivity [5]. But in the general
case, the signal terms of TFD are always positive or zero, and only the interference terms
can be also negative [6, 7]. After a radical smoothing of interference terms by means of a
Fourier transform Fourier energy Fourier energy
X(j) spectrum X(j) 2 cepstrum Cx()averaging over time averaging over time /Cx() Gx()/
2
X(j) TFD t; dtx( )= X xG TQD t dt() (; )=
ambiguity time-frequency time-quefrency
function AX(;) distribution TFDx(t;) distribution TQDx(t;)FT; [AX(; ;)H( ; )]IFTt [x*(t-/2)x(t+/2)]
FTt [x(t)]IFT [log()]
IFT [log ]
Signalx(t)
X(j).X*(j)
Figure 1. Mutual relations between all transformations of a measured signal and distributions in frequency,quefrency, timefrequency, timequefrency and ambiguity domain.
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well chosen smoothing kernel, the negative terms almost vanish and we can consider the
TFD as nearly positive. Therefore, we can replace the TFDXt;o in equation (4) by its
magnitude:
TQDxt; t
Z11
ln TFDxt;oj j expjot do: 7
Using equation (7) instead of equation (4) we can economise a lot of calculating time by
renunciation of complex algebra. The time averaging of TQDXt;o can be done
subsequently with formula (5).
According to equations (4) and (7) for every representant of Cohen class TFD we can
find an appropriate TQD. All these are members of a new class of distributions in the
timequefrency domain.
In the next section, an application of the described new TQD will be shown. We will
detect and identify time-independent (Section 3.1) and time-dependent (Section 3.2)
periodicities of sidebands in the measured acceleration of a gearcase of a passenger car [3].
The TQD proves to be a strong tool for the recognition of toothing faults or damagesduring test runs with fast-changing rotational speed here.
3. APPLICATION OF TQD FOR GEAR DRIVE DIAGNOSTICS
The investigated manual-controlled gearbox of a passenger car as well as the
experiments are described in Part I of our paper [3].
3.1. TQA OF ANGLE-EQUIDISTANT SAMPLED MEAUSURED DATAS
The central part of Fig. 2 shows the contour plot of the CWD of the angle-equidistant
sampled measured signal of structure-borne noise at the gear housing. You can see thetime function in the left part and the ordering spectrum at the bottom of the same figure.
Figure 3 shows the timequefrency distribution TQDXt; t; which was calculated by the
inverse Fourier-transformation of the logarithm of magnitude of CWD according to
equation (7). Some vertical straight lines can be recognised. The dominating ones at the
0.327th and the 0.527th-order quefrency belong to the rotational speed of the central gear
and annulus of the second planetary set, resp. These cepstral lines characterise
corresponding sidebands in the frequency domain and modulations in the time domain.
All these properties are symptoms for existing faults or damages in these gear wheels. This
is in accordance with the statement, which we could make on the basis of ordinary
Fourier- analysis in Part I of our paper [3]. Now it is evident that the TQA is suitable fordiagnosis of gears.
The other vertical straight lines in Fig. 3 are higher and broken rahmonics of the
rotational order of sun gear or annulus resp. The broken rahmonics correspond to higher
harmonics of rotational orders in the ordering spectrum, which normally appear as a
result of the mentioned faults. The higher rahmonics are generated by the numerical
calculation of logarithm of the timefrequency distribution TFDXt;o [4]. They are
numerically generated artefacts.
The ordering cepstrum at the bottom of Fig. 3 was obtained by averaging of the TQD
over time according to equation (5). This averaging could be done without any smearing
of peaks because of the data sampling in equidistant angular intervals. Once more weemphasise, that this cepstrum GXt is not equal to the Fourier- power cepstrum CXt of
signal xt (see Section 2). But nevertheless, the GXt cepstrum can be assessed instead of
the real power cepstrum CXt because of its similarity. Doing this, the appearance of
peaks is a sure indication for periodicities in the appropriate spectra. The cepstral peaks,
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which are caused by faults in sun wheel and annulus, are clearly shown in the ordering
cepstrum GXt in Fig. 3.
As the result of this section we can state that the angle-equidistant sampled signals
seems to be stationary because of the constant angle intervals at the rotating incremental
encoder (they can be influenced only by transient resonances during the start up or run
down). Therefore, the rotational orders of gear wheels keep constant over the time. As a
reflection of this fact, we can consider the perpendicular straight cepstral lines in Fig. 3. Inthe next section, we will analyse the time-equidistant sampled signals. The result will be
inclined cepstral lines in the TQD, whose inclination depends on the changing rate of
rotational speed. So we are able to detect the periodicities of sidebands also in the case of
quickly changing rotational speed.
3.2. TQA OF TIME- EQUIDISTANT SAMPLED MEASURING DATAS
The measured signal is sampled by the impulses of an electronic generator in equal time
intervals. Figure 4 (central part) shows the TFD of one of these signals [3]. Let us repeat
once more: the timefrequency transformation was done with the use of a Cohen class
transformation with a smoothing kernel, which respects the change of rotational speed.Therefore, the frequency lines with the meshing frequency and its sidebands appear nearly
continuously (the discontinuities at the beginning and at the end of the analysed time
interval are typical effects of Cohen class). The measured time function and the frequency
spectrum over the total measuring time interval are also shown at the left and low margins,
Figure 2. CWD with s 0:05 for three revolutions of the angle-equidistant sampled measuring signal of thegearbox (1 and 2 mark individual sidebands).
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resp. From this TFD, the timequefrency transformation was numerically done according
to equation (7). The result is shown in Fig. 5. Once more straight lines are orientated
nearly perpendicularly. They reflect the distance of sidebands in the time-dependent
spectrum (see TFDxt;o in Fig. 4). The dominating ones lie at tC2 % 0:020 s and tA2 %
0:032 s: These quefrencies exactly correspond with time for one revolution of the sun wheel(central gear) and annulus of the second planetary set, resp. Therefore, we can state that
they reflect the sidebands, which belong to the damaged wheels.
Nevertheless, all straight lines in Fig. 5 are slightly inclined to the time axis}this is a
symptom for the change of rotational speed of the damaged (or badly manufactured)
wheels during the measuring time of cca. T% 0:212 s (three revolutions of output shaft).
Because of the insignificance of this inclination, we dared to average the timequefrency
distribution TQDXt; t along the time axis according to equation (5) in order to compare
the result with the ordering cepstrum, obtained by angle-equidistant sampling. Therefore,
we got the appropriate peaks in the cepstral function GXt}see Fig. 5, low part. Here can
we very well recognise the cepstral peaks at quefrencies tC2 % 0:020 s and tA2 % 0:032 s:These are symptoms for faults or damages at the flanks of sun wheel (marked with C) and
the annulus (marked with A). Comparing these peaks with the same peaks in Fig. 3 we
must state that they are some smeared over because of the weak unstationarity of the
sideband distance, of course. But nevertheless a quantitative analysis is able now: by
Figure 3. Timequefrency distribution TQDxt; t of the angle-equidistant sampled signal (above) and theaveraged ordering cepstrum Gxt (below).
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means of cursor at certain time moments from the contour plot or also by means of the
averaged cepstrum at the bottom margin. A better recognition could be reached by
averaging along the inclined line of sideband quefrencies.
The apperance of these peaks in the averaged TQD is a clear advantage over the
ordinary Fourier-cepstrum (in which these peaks were not recognizable because of thechanging rotational speed of the damaged wheels}see Part I of our paper [3]).
Once more we must emphasise that the use of the special smoothing kernel with
consideration of the sweeping rotational speed is an indispensable supposition for getting
the meaningful TQD according to Fig. 5. For underlining this statement, we calculated the
ordinary CWD (its kernel does not respect the time dependence of the rotational speed
[57]) of the same structure-borne noise during the start up. This CWD (s 0:05) is
shown in Fig. 6. This plot is already analysed in Part I of our paper [3]. Figure 7 shows the
belonging to TQD. All expected side-band-related cepstral lines (in the TQD) and cepstral
peaks (in the averaged cepstrum at the bottom of Fig. 7) are vanished}a consequence of
the non-observance of change of rotational speed. This is the promised pragmatic evidenceof the unsuitability of the CWD for the recognition of chirps (see Section 5.2 of Part I [3]).
Already the analysis of the CWD-plot in Fig. 6 mislead to an undervaluation of the
sidebands because of their breaks along the time axis. The final result is a false diagnosis of
the technical state of the monitored gear drive.
Figure 4. The timefrequency distribution TFDx
t;o of the time-equidistant sampled measuring signal ofgearbox during linear start up (weighted with the acceleration-respecting kernel; 1 and 2 mark individualsidebands).
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4. SUMMARY
The target of paper in hand is the detection and diagnosis of faults and damages at gear
drives under special operational conditions and limited hardware conditions. In contrast
to the already known and applied approaches [811], we must reach this target by
measurement under strong unsteady operational conditions, e.g. during a fast start up or
run down. As tools we used the conventional Fourier and cepstral transformations as wellas the timefrequency analysis in the special form of timefrequency distributions of
Cohen class with different successful results.
While in Part I [3], a new smoothing kernel was created, which respects the change of
rotational speed during the measurement; in Part II, we used the timequefrency
Figure 5. Timequefrency distribution TQDxt; t of the time-equidistant sampled signal, which wascalculated outgoing from the timefrequency distribution TFDxt;o with the special smoothing kernelHCn; t under consideration of angular acceleration (above) and the averaged cepstrum Gxt (below).
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distribution on the basis of the time-dependent cepstral analysis as a improved signal
transformation in the timequefrency domain.
In connection with the mentioned special smoothing kernel distribution and with the
given-by Cohen class analysis premise of a good frequency resolution and continously
marked sideband lines, the timequefrency allows the detection and identification of faultsand damages, e.g. at tooth flanks of gear wheels, also in the case of instationary rotational
speed and without implementation of any incremental sender for rotational angle [1]. This
was verified at a planetary gear of passenger car with in-advance-known faults as a case
study.
The proposed method for gear-drive monitoring and diagnosis of faults and damages
requires much less hardware components compared with the traditional approaches. The
disadvantage is a higher demand in numerical computation.
ACKNOWLEDGEMENTSThis research was supported by the Deutscher Akademischer Austauschdienst (DAAD)
by awarding the postdoc-scholarship No. A/97/13553. An additional subvention was given
by the State Ministry for Sciences and Art of Saxony. We thank Mr. D. Lieske for making
the measurements.
Figure 6. CWD with s 0:05 of the time-equidistant sampled measuring signal of gearbox during linear start
up (1 and 2 mark individual sidebands).
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